Introduction to Mechanical Behavior of Biological Materials Ozkaya and Nordin Chapter 7, pages 127-151 Chapter 8, pages 173-194 Outline Modes of loading Internal forces and moments Stiffness of a structure under uniaial loading Stress and strain due to aial loading Elastic modulus of a material Definitions from the stress-strain diagram Shear stress and strain Stresses due to bending and torsion Composite structures: springs in parallel and in series KIN 201 2007-1 Stephen Robinovitch, h.d. 1 2 orces cause motions and deformations force motion deformation if the net eternal force or moment applied to a body is not zero, it will undergo gross motion if a body is subjected to eternal forces or moments but remains in static equilibrium, there will be local changes in the shape of the body, or deformations in our previous treatment of static equilibrium, we assumed that bodies were infinitely stiff and could not deform we now rela this assumption to eamine the force-deformation behavior of biological materials 3 There are five primary modes of loading compression and tension are both aial loading by convention, compressive loads are negative, and tensile loads are positive in reality, we rarely have pure aial loading, pure bending, or pure torsion, but instead combined loading 4
Each mode of loading creates a distinct pattern of internal stress tension compression shear! =! = - bending M M! = My I torsion T T! =! = Ty J 5 Internal forces and moments for a structure in equilibrium, eternal forces are balanced by internal forces and moments we usually resolve internal forces and moments into components parallel and orthogonal to the long ais of the structure 6 Internal forces and moments (cont.) : aial force (tends to elongate the structure); we shall simply use y, z : shear force (tends to shear the structure apart); we shall simply use V M : twisting moment or torque (tends to twist the structure about its long ais); we shall simply use T M y, M z : bending moments (about the structure s short aes); we shall simply use M y V M 7 Eample: internal forces and moments During single legged stance, the proimal femur is loaded as shown. Determine the forces and moments acting at the femoral neck cross-section -.! J b e c a Section -: D c a b 8
Eample: internal forces and moments (cont.)! = 0 : = J cos"! y = 0 : V = J sin"! M a = 0 : M = " J sin# $ e Tensile loads cause a body to stretch Consider a structure of original length and crosssectional area, that is loaded by two equal and opposite forces of magnitude we will use the Greek letter delta (") to represent the change in length (or deflection) of the structure after application of the forces + " 9 10 Stiffness under aial loading stiffness is the slope of the load-deformation curve we shall denote stiffness with the letter k units for stiffness are N/m stiffness is affected by the structure s geometry (length and crosssectional area), and is therefore a structural property load (N) stiffness k deflection " (m) 11 Biological structures have nonlinear stiffness if a structure is made from a linear material, the stiffness will be constant (regardless of geometry) if the structure is made from a nonlinear material, the stiffness will vary with the load a steel spring has a linear stiffness biological tissues have nonlinear stiffness linear nonlinear constant stiffness k varying stiffness k " " 12
Stiffness depends on geometry and material ial stiffness increases with increasing cross-sectional area, and decreases with increasing length 13 Modulus does not depend on geometry if we divide by the crosssectional area, and " by the original length, the forcedeflection curves collapse onto one common trace the slope of this trace is the modulus of elasticity E (or Young s modulus) Young s modulus does not depend on geometry. It is a material property rather than a structural property the units of E are [N/m 2 ], often called ascals, a % %!! 14 Definition of aial stress and strain the ratio / is the stress " under aial loading; the units of stress are [N/m 2 ] or pascals [a] the ratio "/ is the strain # under aial loading; strain is dimensionless although it is sometimes referred to as having units of [m/m] XI STRESS! = [a] % XI STRIN! = " [m/m] 15 Modulus of elasticity is a material property The modulus of elasticity does not depend on geometry; it is a material property rather than a structural property force (N) M structural slope = stiffness k (N/m) deflection! (m) stress # (M/m 2 or a) (-sectional area) E (modulus) l material slope = modulus E (N/m 2 or a) strain $ (percent)! =! = "l l 16
Hooke s aw for aial loading The constitutive relationship that defines a material s stress-strain behaviour (in the elastic region) is referred to as Hooke s aw. or aial loading, Hooke s law is: This can be epressed as:! = E! = E" ("E" or " E" formula) 17 Stiffness under uniaial loading By definition, stiffness is the ratio of load divided by deflection. Based on Hooke s aw, the stiffness of a structure under aial loading is: k =! = E [N/m] structural stiffness under aial loading +" 18 Eample: stress and strain under aial loading section of tendon having a cross-sectional area of 0.5 cm 2 is subjected to a tension test. Before applying any load, the gage length (distance between and B) is 30 mm. fter applying a tensile load of 1000 N, the distance between and B increases to 31.5 mm. Determine the tensile stress #, the tensile strain $, and the modulus of elasticity E. 19 Eample: stress and strain under aial loading (cont.)! = = 1000 0.00005 = 2 10 7 a = 20 Ma " = # = (0.0315-0.030) 0.030 E =! " = 2 107 0.05 = 400 Ma = 0.05 m/m 20
Young s modulus of various orthopaedic biomaterials 21 Definitions related to the stressstrain diagram energy to failure yield strength ultimate strength modulus ultimate strain yield strain elastic limit toughness 6 1 2 3 5 4 1 6 stress # (ka) 2 1 ESTIC 3 STIC 6 (area under curve) 4 5 strain $ (m/m) rupture Stresses due to bending the aial stress due to bending is given by :! b = M b y I a where M b = " is the bending moment; y is the distance from the neutral ais; and I a = # y 2 d is the area moment of inertia (different than the mass moment of inertia neutral ais (# b = 0) tension y compression # r 2 dm) 23 symmetric structures have a directional dependency to stiffness under bending. or the same applied bending moment, in what orientation is the beam stiffer (i.e., will deform less)? What is the reason for the difference? 24
rea moment of inertia (MOI) for common cross-sections or a rectangular cross section, area MOI varies with the cube of height, but only the first power of width. or a circular section, area MOI varies with radius to the 4th power. 25 rea moment of inertia, a purely geometrical parameter, determines aial stress due to bending t what location are the stresses greatest in beam (1)? Is the same true for beam (2)? What theoretical advantages are provided by the beam (2) design? (1) (2) 26 leural rigidity is the product EI Stress does not depend on modulus :! = M by I zz. But strain depends on modulus : " = M by EI zz. The product EI zz is called the "fleural rigidity." Can you understand the differences in fleural rigidity associated with the different fracture fiation plate designs on the right? 27 Shear Stress and Strain Shear stress causes angular deformation of a material, according to Hooke s aw for shear stress and strain:! y = G" y where G is the shear modulus (in a), & y is the shear stress in the y plane (in a), and ' y is the shear strain in the y plane (in radians). 28
Tension and compression eist in a body subject to pure shear loading Shear stresses and strains due to torsion lanes oriented 45 deg to the direction of loading will have tensile or compressive stress. These normal stresses will be equal in magnitude to the shear stress in planes parallel to the loading. 29 Shear stresses due to torsion are :! = Tr J where J is the polar moment of inertia " r 2 d. Corresponding shear strain are : # = Tr JG. The product JG is called the "torsional rigidity." 30 Shear stresses due to torsion Why are shear stresses in the tibia due to torsion lower in section than section B, even though the thickness of the corte is much greater at B? B 31 Mechanics of the spiral fracture s we shall discuss later in the course, bone is weaker in tension than in compression or shear. Spiral fractures due to torsional loading therefore initiate and propagate 45 deg to the long ais of the bone, where tensile stress is maimum. 32
Combined loading: stresses due to aial and bending loads What is the distribution of stress at section - due to the force? a tension! bending = M I compression compression! aial = " y tension M b = a compression! total = # a I " 1 % $ & 33 Composite structures: springs in parallel (load sharing) each spring element has the same deflection the force in each spring element is not the same the total effective stiffness (k total ), and the total deflection of the system, is dominated by the element with the highest stiffness k b k a k total = k a + k b Composite structures: springs in series (load transfer) each spring element supports the same force the deflection in each spring element is not the same the total effective stiffness (k total ), and the total deflection of the system is dominated by the element with the lowest stiffness k total = k a k b k ak b k a + k b In series or parallel? running shoe and ground right and left upper etremity during push up ankle, knee, and hip joint during single-legged stance sarcomeres in muscle articular cartilage and subchondral bone in knee joint SERIES RE k a k b k b k a 36
Review questions what are the five modes of loading, and what patterns of stress do each of these create? what is the difference between shear force and aial force? between a twisting moment and a bending moment? what is the difference between a linear material and a nonlinear material? for uniaial loading, how do you convert force-deflection data to stress-strain data? what parameters affect the stiffness of a structure under uniaial loading? what is the 'E" formula for deflection of a structure under uniaial loading? what is the formula for stress due to an applied bending moment? how do we define the effective stiffness of springs in parallel? of springs in series? what is definition of yield stress? of failure stress? 37